2.2 DATOS GENERALES DEL INSTITUTO NACIONAL AUTÓNOMO
2.2.1 Historia 42
To better understand the performance of the rough-minus-smooth strategies, in this
section we add controls for additional factors. We first discuss factors that might
influence performance and then evaluate their impact using two methods — double
sorts and [42] regressions.
3.4.1
Liquidity
We observed previously that in Table 3.3 the average market cap across the five quin-
tiles increases with H: rougher stocks tends to be smaller on average. This pattern
suggests the possibility that roughness may reflect lower liquidity and therefore that
a rough-minus-smooth strategy earns an illiquidity premium. This possibility is tem-
pered by the fact that the stocks that pass the filters for calculating implied roughness
are larger, on average, than those that do not. The question therefore requires a more
systemic investigation.
A connection between realized roughness and liquidity was noted in an early
version of [16], but it was removed from subsequent versions of that paper. [16]
compare estimates of realized roughness with daily volume of trading in a stock.
In addition to trading volume, we consider the widely-used [4] illiquidity measure.
The Amihud measure for a single stock in a single month sums the absolute values
of the daily returns and divides the sum by the dollar volume for the month. Larger
values of the Amihud measure are interpreted as indicating lower liquidity, whereas
larger values of trading volume are associated with greater liquidity.
Figures 3.5 and 3.6 compare, respectively, realized and implied estimates of H
with the log of the Amihud measure and log daily volume. Each dot in the figure
corresponds to a single stock in a single month. Consistent with the earlier version
Figure 3.5: Realized roughness and liquidity. The figures plot realized roughness against the log of the Amihud illiquidity measure (top) and log daily volume (bottom). Each point shows a single stock in a single month.
Consistent with this pattern, we find a negative correlation (−0.46) between realized
H and the log Amihud measure.
The results using implied roughness in Figure 3.6 are qualitatively similar but not
as strong. The correlation between implied H and log daily volume is 0.28, and the
correlation with the log Amihud measure is −0.40.
Beyond these empirical patterns, a potential link between roughness and liquidity
is interesting because of efforts to explain realized roughness through market mi-
crostructure; see [38] and [62]. However, the explanations developed to date are
highly stylized, and they do not make clear predictions about whether greater rough-
ness should be associated the more or less liquidity.4
4According to Mathieu Rosenbaum (personal communication), [62] implies a longer transient
Figure 3.6: Implied roughness and liquidity. The figures plot implied roughness against the log of the Amihud illiquidity measure (top) and log daily volume (bottom). Each point shows a single stock in a single month.
3.4.2
Implied Volatility and Skewness
Implied roughness is a relatively complex feature of a stock’s implied volatility surface,
involving differences in implied volatilities across both strikes and maturities. To try
to isolate the source of alpha in the implied rough-minus-smooth strategy, we will
therefore control for more basic features — the level of the ATM implied volatility
and the shape of implied volatility skew.
Several authors (particularly [34] and [91]) have documented predictability in
stock returns using measures of implied volatility and skewness. A fast decay in the
ATM skew (low implied H) is potentially associated with high degree of near-term
skewness or implied volatility. We therefore control for these factors.
As our measure of ATM implied volatility, we use the implied volatility for a
surface data set from OptionMetrics. We denote this by σCall
1m (KS = 1). Similar to
[91], we use as our measure of implied volatility skew
XZZ-skew =σ1P utm (K S = 0.9)−σ Call 1m ( K S = 1), (3.11)
the difference between the one-month implied volatility for a put with moneyness
closest to 0.9 and the one-month implied volatility for a call with strike closest to the
spot price.
[91] find that larger values of their skew measure predict lower stock returns in
the cross section, a pattern that we find holds up as well using more recent data
and a slightly different skew measure. Interestingly, this effect appears to run in the
opposite direction of what we find using implied roughness. A smaller implied H
indicates a faster decay of the ATM skew. If this indicates a higher initial value of
the ATM skew, then the finding of [91] would suggest that stocks with smaller implied
H have lower stock returns, yet we find exactly the opposite. This suggests that the
performance of the rough-minus-smooth strategy is not explained by the XZZ-skew,
a hypothesis we will check in the next sections.
3.4.3
Double Sorts
To control for factors like liquidity or skewness that might influence the returns on
our roughness quintile portfolios, in this section, we apply a standard double-sorting
procedure.
Suppose, for example, that we want to control for illiquidity, using the Amihud
measure. For each month, we proceed as follows. We sort stocks into deciles according
to the Amihud measure. Within each of these illiquidity deciles, we sort stocks by
roughness (realized or implied). We then take the roughest quintile from each of
portfolio by grouping all stocks that are in the smoothest quintile of any of the
illiquidity deciles.
Under this construction, all levels of illiquidity are represented in the rough and
smooth portfolios, so the performance of the rough-minus-smooth strategy should be
unaffected by illiquidity: we have hedged out illiquidity. We sort into ten portfolios
based on illiquidity in the first step in order to achieve a better balance of the con-
ditioning factor between our controlled rough portfolio and smooth portfolio. The
same procedure allows us to hedge out the effect of any other factor by first sorting
on that factor.
We apply double sorts that condition on the following variables, one at a time:
◦ Average daily volume for each stock;
◦ The Amihud illiquidity measure;
◦ Turnover, measured as a stock’s monthly trading volume divided by the average shares outstanding of that stock during the month;
◦ ATM implied volatility, as measured by the implied volatility for a 30-day option with strike closest to spot price;
◦ XZZ-skew, as defined in (3.11);
◦ Size (as measured by log market cap), book-to-market, and trailing 12-month return.
Table 3.5 shows the performance of the rough-minus-smooth strategy based on
implied roughness after controlling for each of these factors through double sorts. The
table shows average returns and alphas using either FF3-Mom or FF5-Mom factor
models.
The first three rows of the table consider liquidity measures. Sorting first on
the profitability of the strategy. Some reduction in performance is to be expected,
given the correlation we documented in Section 3.4.1 between implied roughness and
these measures. But the profitability of the strategy remains significant, particularly
as measured by alpha relative to the Fama-French 5-factor with momentum, ranging
from 3.1% to 5.4% per year, depending on the measure used, witht-statistics ranging
from 2.0 to 3.0. Controlling for turnover actually increases the mean return of the
strategy, with average monthly returns of 0.54%, and increases the t-statistics to
around 4.0. In short, liquidity by itself cannot account for the performance of the
rough-minus-smooth strategy.
The next two rows of the table control for implied volatility and the ATM skew.
Controlling for ATM implied volatility improves the average return and alphas to 7%,
except for the FF5Mom alpha, which decreases a bit to 4.9% annually. Controlling
for the XZZ-skew measure of [91] has only a small effect on the average return, alphas
andt-statistics, and all alphas remain statistically significant. Thus, these well-known
features of the implied volatility surface — the level of ATM volatility and skewness in
implied volatility — cannot account for the performance of the rough-minus-smooth
strategy.
The last three factors in the table serve as robustness checks. Sorting on size, book-
to-market, and trailing returns may slightly reduce the performance of the strategy
but does not eliminate — and may even strengthen — statistical significance.
Table 3.6 shows corresponding results based on realized roughness, using the full
universe of stocks (Panel A) or the implied universe (Panel B). Here we find that con-
trolling for liquidity (through average daily volume or the Amihud measure) removes
the significance of returns and alphas of the rough-minus-smooth strategy. Control-
ling for size does as well in Panel A. These results suggest a strong association between
realized roughness and illiquidity. In contrast, controlling for implied volatility and
Conditioning Variable Mean Return CAPM Alpha FF3Mom Alpha FF5Mom Alpha
Average Daily Volume 0.23* 0.21 0.23* 0.26**
[1.84] [1.60] [1.81] [2.01]
Average Daily Amihud 0.45*** 0.46*** 0.46*** 0.40***
[3.23] [3.31] [3.21] [2.65]
Turnover 0.54*** 0.53*** 0.52*** 0.45***
[3.96] [3.90] [3.53] [2.98]
XZZ Skew 0.46*** 0.44** 0.49*** 0.45***
[2.79] [2.54] [2.96] [2.61]
ATM Implied Volatility 0.59*** 0.59*** 0.59*** 0.41**
[3.54] [3.51] [3.32] [2.28] Size 0.41*** 0.42*** 0.46*** 0.42*** [3.24] [3.31] [3.53] [3.16] Book-to-Market 0.34** 0.32** 0.36** 0.35** [2.44] [2.17] [2.57] [2.34] 12-Month Return 0.45*** 0.43*** 0.43*** 0.48*** [3.29] [3.06] [3.17] [3.38]
Table 3.5: Performance of rough-minus-smooth portfolios using implied roughness, constructed through double sorts on various factors, for the period Jan 2000 through Jun 2016. Mean return and alphas are monthly values in percent. Numbers in brackets are t-statistics based on Newey-West standard errors.
cates that the effect of roughness, whether realized or implied, is not already reflected
in the ATM volatility or the ATM skew.
3.4.4
Fama-MacBeth Regressions
To further investigate whether the performance of the rough-minus-smooth strategy
is explained by other factors, we run regressions based on the specification
Reti,t =b0t+b1tHi,t+b02tCON T ROLSi,t−1+ei,t, (3.12)
where Reti,t is the return of stock i in month t; Hi,t is either realized or implied
roughness of stock i in month t; CON T ROLSi,t−1 is a vector of controls; and the
ei,t are error terms. We estimate coefficients and their standard errors through [42]
regressions: in each month t, we run cross-sectional regressions to estimate b0t, b1t,
and b2t; we then take the time-series averages of these regression coefficients and use
their time-series variation to estimate standard errors. Compared to the double sorts
Conditioning Variable Mean Return CAPM Alpha FF3Mom Alpha FF5Mom Alpha PANEL A: Full Universe
Average Daily Volume 0.14 0.15 0.14 0.05
[1.47] [1.63] [1.51] [0.60]
Average Daily Amihud 0.12 0.17 0.12 -0.05
[0.94] [1.37] [1.03] [-0.45]
Turnover 0.38*** 0.38** 0.33** 0.20
[2.60] [2.49] [2.34] [1.46]
XZZ Skew 0.54*** 0.57*** 0.53*** 0.30*
[3.10] [3.22] [3.36] [1.95]
ATM Implied Volatility 0.54*** 0.56*** 0.59*** 0.41**
[2.81] [2.92] [3.17] [2.15] Size 0.12 0.18 0.13 -0.04 [0.90] [1.47] [1.27] [-0.37] Book-to-Market 0.35*** 0.38*** 0.35*** 0.20 [2.63] [2.90] [2.73] [1.55] 12-Month Return 0.51*** 0.54*** 0.50*** 0.37** [3.06] [3.12] [3.19] [2.34]
PANEL B: Implied Universe
Average Daily Volume 0.22 0.25 0.21 0.08
[1.43] [1.64] [1.48] [0.56]
Average Daily Amihud 0.40* 0.47** 0.41** 0.22
[1.94] [2.31] [2.40] [1.35]
Turnover 0.60*** 0.61*** 0.63*** 0.53***
[3.13] [3.09] [3.38] [2.79]
XZZ Skew 0.49** 0.53** 0.51*** 0.27
[2.41] [2.51] [2.88] [1.54]
ATM Implied Volatility 0.62*** 0.63*** 0.62*** 0.43*
[2.81] [2.77] [2.88] [1.95] Size 0.49** 0.55*** 0.53*** 0.35** [2.41] [2.70] [3.13] [2.10] Book-to-Market 0.31* 0.34** 0.33** 0.14 [1.83] [2.05] [2.05] [0.88] 12-Month Return 0.55*** 0.60*** 0.57*** 0.41** [2.96] [3.08] [3.47] [2.47]
Table 3.6: Performance of rough-minus-smooth portfolios using realized roughness, constructed through double sorts on various factors, for the period Jan 2000 through Jun 2016. Mean return and alphas are monthly values in percent. Numbers in brackets are t-statistics based on Newey-West standard errors.
inclusion of multiple controls, but they have the disadvantage of imposing linearity
on the relationship between returns and controls.
An alternative approach would be to run a panel regression to estimate (3.12)
with no dependence on t in the coefficients. Since we are mainly interested in the
cross-sectional relationship between roughness and returns, we would include month
fixed-effects; and since monthly returns have very low autocorrelation, we would es-
timate standard errors clustered by month, following [75]. However, as also discussed
in [75], Section 3, Fama-MacBeth standard errors are more accurate than panel re-
gressions with clustered standard errors under two conditions that are appropriate to
our setting: (1) the main source of dependence in error terms comes from time effects
(correlations in returns of different stocks in the same month); and (2) the number of
time periods (201 months) is not very large compared with the number of stocks per
month (up to 1108 stocks per month in the implied universe and 3577 per month for
the full universe). The dependence in (1) is dealt with effectively by Fama-MacBeth
regressions. The values in (2) would require the estimation of a very large covariance
matrix between different stocks based on limited data in order to cluster by time. In
light of these considerations, we use Fama-MacBeth regressions.
Table 3.7 shows the results. Panel A tests implied H; Panel B test realizedH on
the implied universe; and Panel C tests the realizedH on the full universe of stocks.
Each panel shows two regressions, one including only the corresponding roughness
measure, and one including multiple controls. All explanatory variables have been
standardized (cross-sectionally in each month) to make the coefficients comparable.
Returns are in decimals, so a return of 5% is recorded as 0.05.
Panel A confirms the negative relationship between returns and implied H; in-
cluding controls increases the magnitude and significance of the coefficient. Panel B
shows that realized H has a significant relationship with returns when restricted to
find no significant relationship between realizedH and returns on the full universe of
stocks, with or without controls. Interestingly, our results confirm a strong negative
relationship between returns and the skewness measure of [91], while also showing in
Panel A that this control does not explain the effectiveness of implied roughness.
Our controls include return volatility and implied volatility, so the regressions
in Table 3.7 also control for the volatility risk premium ([27]) measured as the dif-
ference between implied and realized volatility. In particular, Panel A shows that
the profitability of the implied strategy cannot be attributed to the volatility risk
PANEL A PANEL B PANEL C
Variable Reg 1 Reg 2 Reg 3 Reg 4 Reg 5 Reg 6
Intercept 0.0043 0.0046 0.0043 0.0046 0.0088* 0.0078 [0.83] [0.91] [0.83] [0.90] [1.66] [1.49] Implied H -0.0010** -0.0014*** [-2.04] [-3.43] Realized H -0.0015** -0.0003 -0.0003 -0.0002 [-2.10] [-0.68] [-0.51] [-0.56] XZZ Skew -0.0034*** -0.0034*** -0.0036*** [-5.30] [-5.20] [-6.56] ATM volatilities -0.0063*** -0.0062** -0.0047** [-2.62] [-2.56] [-2.25]
Log Option Volume -0.0033* -0.0034* -0.0015
[-1.85] [-1.91] [-1.44]
Log Option Open Interest 0.0025 0.0024 -0.0012
[1.58] [1.53] [-1.22]
Log Stock$Volume 0.0044 0.0046 0.0006
[1.38] [1.45] [0.25]
Log Stock Volume 0.0019 0.0018 0.0061***
[1.06] [1.03] [3.58] Turnover -0.0019 -0.0020 -0.0027** [-1.47] [-1.54] [-2.51] Book-to-Market -0.0003 -0.0002 -0.0010 [-0.29] [-0.21] [-0.36] Log Size -0.0095*** -0.0094*** -0.0079*** [-2.89] [-2.84] [-3.01] Past 6M Return -0.0006 -0.0007 -0.0007 [-0.49] [-0.56] [-0.59] Past 12M Return 0.0010 0.0011 0.0010 [0.93] [0.98] [1.01]
Past Return Volatility -0.0024* -0.0024 -0.0043***
[-1.65] [-1.63] [-2.76]
Past Return Skew -0.0005 -0.0004 -0.0002
[-0.95] [-0.88] [-0.60]
Adj. R2 0.29% 13.15% 0.46% 13.18% 0.14% 9.21%
Table 3.7: Fama-MacBeth regression results. Panel A, B, C each have two regression results, one with only one regressor (either implied or realized H) and the other including a complete set of controls. Panel A shows results for implied H. Panel B presents results for realized H on the implied universe. Panel C uses realized H and the unrestricted universe. Numbers in brackets are t-statistics based on Newey-West standard errors.